Cambrian combinatorics on quiver representations (type A)

Abstract

This paper presents a geometric model of the Auslander-Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We show that its endomorphism algebra is a tilted algebra of type A. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice.

arXiv comment28 pages, 16 figures. Comments are welcome

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• Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf{E}$-clusters
• A geometric realization of socle-projective categories for posets of type $\mathbb{A}$
• $c$-Birkhoff polytopes